Why can't you prove triangles congruent by using side-side-angle?
Consider the two triangles.
The red sides are congruent, the blue sides are congruent, and the green angles are congruent.
However, even though the triangles would seem to be side-side-angle congruent, they're not. Every side-side-angle combination can be used to describe two different triangles.
(The one exception is if the angle is a right angle, for which only one triangle can be drawn. This special case of side-side-angle congruence is called the hypotenuse leg theorem and is valid.)